The Yankee center fielder homered in his first at-bat in the second inning, doubled in the fourth, singled in the fifth and tripled in the ninth inning off Sox pitcher Scott Linebrink.
"I feel really happy about it, but I'm more happy because we won the game," Cabrera told reporters through a translator.
"During the beginning of the game, I didn't think about it. But when I got a double and a home run and a base hit I knew a triple could do it."
Cabrera completed the cycle in the ninth inning after he hit a line-drive triple that bounced off the wall in right field.
"Everyone was pulling for him," Yankee captain Derek Jeter said. "It's difficult. You can't just go up there and try to hit a triple. He hit it in the perfect spot. It looked like he got a nice little kick to it. I think everyone in the dugout was running right there with him."
The last Yankee to hit for the cycle was Tony Fernandez, who accomplished the feat against the Oakland A's in 1995. The last cycle hit against the Sox was Carlos Gomez, who did it for the Twins in 2008 at U.S. Cellular Field.